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1. ¤Ø³ÊÁºÑµÔ¾ÔàÈɢͧàÅ¢ 9
(1)¶éÒ·èÒ¹¹ÓàÅ¢ 9 令ٳ¡Ñº¨Ó¹Ç¹ã´æ¤ÓµÍº·Õèä´é ¹ÓÁҺǡ¡Ñ¹¨Ðä´éàÅ¢ 9 àÊÁÍ àªè¹
9x2=18 ¹Ó¤ÓµÍº·Õèä´é¤×Í 18 ÁҺǡ¡Ñ¹ä´é 1+8=9
9x3=27 ¹Ó¤ÓµÍº·Õèä´é¤×Í 27 ÁҺǡ¡Ñ¹ä´é 2+7=9
9x4=36 ¹Ó¤ÓµÍº·Õèä´é¤×Í 36 ÁҺǡ¡Ñ¹ä´é 3+6=9
9x8=72 ¹Ó¤ÓµÍº·Õèä´é¤×Í 72 ÁҺǡ¡Ñ¹ä´é 7+2=9
(2)·èÒ¹Åͧ¡µÑÇÍÂèÒ§àÅ¢¨Ó¹Ç¹Ë¹Öè§àªè¹ 987 ËÒ¡ÊÅѺ·Õè¨Ðä´é 978 ËÃ×Í 798 ¨Ò¡¹Ñé¹¹ÓàÅ¢µÑǹéÍÂä»ÅºÍÍ¡¨Ò¡àÅ¢µÑÇÁÒ¡
¼ÅÅѾ¸ì·Õèä´éàÁ×èÍËÒôéÇ 9 ¨ÐŧµÑÇàÊÁÍ àªè¹

879_
798
081
¼ÅÅѾ¸ì 81 ¨ÐËÒôéÇ 9 ŧµÑÇàÊÁÍ
(3)·èÒ¹Åͧ¡µÑÇÍÂèÒ§àÅ¢¨Ó¹Ç¹Ë¹Öè§àªè¹ 987 ¹ÓàÅ¢áµèÅÐËÅÑ¡ÁҺǡ¡Ñ¹ä´é 9+8+7 = 24 áÅéǹÓä»Åº ¡ÑºµÑÇàÅ¢à´ÔÁ¨Ðä´é
987-24=963 ·èÒ¹¨Ð¾ºÇèÒ¼ÅÅѾ¸ì·Õèä´é¤×Í 963 ¨ÐËÒôéÇÂ9 ŧµÑÇàÊÁÍ ¶éÒäÁèàª×èÍÅͧ¡µÑÇÍÂèÒ§Í×è¹´Ù¡çä´é

(4)àÅ¢ 9 ÂѧÂդسÊÁºÑµÔ¾ÔàÈÉÍÕ¡àªè¹
987654321x9 = 8888888889
987654321x18 = 1777777778
987654321x27 = 2666666667
987654321x36 = 3555555556
987654321x45 = 4444444445
987654321x54 = 5333333334
987654321x63 = 6222222223
987654321x72 = 7111111112
987654321x81 = 8000000001
(5)á¤è¹ÕéÂѧäÁè¾ÍàÅ¢ 9 ÂѧÁդسÊÁºÑµÔ¾ÔàÈɴѧ¹Õé
1x9+2 = 11
12x9+3 = 111
123x9+4 = 1111
1234x9+5 = 11111
12345x9+6 = 111111
(6)àÅ¢ 9 ÂѧÁդسÊÁºÑµÔ´Ñ§ÃٻẺ¹Õé´éÇ ¤×Í
1x8+1 = 9
12x8+2 = 98
123x8+3 = 987
1234x8+4 = 9876
12345x8+5 = 98765

ÊÃØ» ¨Ò¡¤Ø³ÊÁºÑµÔ¾ÔàÈɢͧàÅ¢ 9 ·Õè¡ÅèÒÇÁÒáÅéÇàÃÒÊÒÁÒöãªé»ÃÐ⪹ì㹡ÒäԴàÅ¢ãËéàÃçÇ¢Öé¹â´ÂÁÕËÅÑ¡´Ñ§¹Õé
àÁ×è͵éͧ¡ÒäٳàŢ㴴éÇ 9 àªè¹ 8x9 ËÃ×Í 987x9 àÃÒÁÕÇÔ¸Õ¤Ô´ÅÑ´´Ñ§¹Õé
(1)Å´àÅ¢·Õè¨Ð¤Ù³¡Ñº 9 Å§ä» 1 ä´éà·èÒäËÃÒ à¢Õ¹ãÊè㹡ÃдÒɤӵͺ
(2)ËÒàÅ¢·Õè¹ÓÁҺǡ¡ÑºàÅ¢·ÕèŴŧ 1 áÅéǹÕéãËéä´é 9 àµÔÁµèÍ·éÒ ¡ç¨Ðà»ç¹¤ÓµÍº àªè¹
8x9 = ? ÇÔ¸Õ¤Ô´
(1)Å´ 8 ŧ 1 ä´é 7 à¢Õ¹àÅ¢ 7 äÇé㹤ӵͺ
(2)ËÒàÅ¢ÁҺǡ¡Ñº 7 ä´é 9 àÅ¢¹Ñ鹤×Í 2 áÅéÇà¢Õ¹ 2 µèÍ ·éÒ¤ӵͺ ¨Ðä´é 72
à¾ÃÒЩйÑé¹ 8x9 = 72
ÍÕ¡µÑÇÍÂèҧ˹Öè§
54x99 = ? ÇÔ¸Õ¤Ô´
(1) Å´ 54 ŧ 1 ä´é 53 à¢Õ¹àÅ¢ 53 äÇé㹤ӵͺ
(2)ËÒàÅ¢ÁҺǡ¡Ñº 5 ãËéä´é 9 àÅ¢¹Ñ鹤×Í 4 à¢Õ¹ 4 µèÍ ·éÒ¨Ðä´é 534 ÂѧäÁ診¤Ð
ËÒàÅ¢ºÇ¡¡Ñº 3 ä´é 9 àÅ¢¹Ñ鹤×Í 6 à¢Õ¹ 6 µèÍ·éÒ ¨Ðä´é5346 à»ç¹¤ÓµÍº
à¾ÃÒЩйÑé¹ 54x99 = 5346
ã¤ÃÁÕÍÐäÃà¡ÕèÂǡѺàÅ¢ 9 ·Õèʹء à¢Õ¹ Êè§ ÁÒä´é¹Ð¤Ð




2. ¨µØÃÑÊÁËÑȨÃÃÂì

¤ÓÇèÒ"¨µØÃÑÊ" ¤×Í ÊÕèàËÅÕèÂÁ¨µØÃÑÊÁËÑȨÃÃÂìµÃ§ä˹ àªè¹
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753
618
¢Ñ鹵͹ :
(1)ºÇ¡àÅ¢á¹Ç¹Í¹ã¹áµèÅÐá¶Çà¢éÒ´éÇ¡ѹ ¨Ðà·èҡѹ
2+9+4 = 15,7+5+3 = 15,6+1+8 = 15
(2)ºÇ¡àÅ¢ã¹á¹ÇµÑéÇã¹áµÅÐËÅÑ¡à¢éÒ´éÇ¡ѹ¨Ðà·èҡѹ
2+7+6 = 15,9+5+1 = 15,4+3+8 = 15
(3)ºÇ¡àÅ¢µÒÁá¹Ç·Ðá§à¢Ò´éÇ¡ѹ(¨Ò¡ÁØÁ˹Öè§ä»ÍÕ¡ÁØÁ˹Öè§)
2+5+8 = 15, 6+5+4 = 15
ÊÃØ»¤ÇÒÁÁËÑȨÃÃÂìÍÂÙè·Õè äÁèÇèÒàÃҨкǡàÅ¢·Ò§ã´¡çµÒÁàÃÒ¨Ðä´é¼ÅÅѾ¸ìà·èҡѹàÊÁÍ
µÑÇÍÂèÒ§¨µØÃÑÊÁËÑȨÃÃÂìÃÙ»Í×è¹æ àªè¹
672
159
834
438
951
276
816
357
492
492
357
816


·èÒ¹ÅͧÊÁÁصԵÑÇàÅ¢Í×è¹ áÅéÇà»ç¹ä»µÒÁà§×è͹䢢éÒ§µé¹ Êè§ÁÒÃèÇÁʹء¡ÑºàÃÒä´é

¨µØÃÑÊÁËÑȨÃÃÂìªÑé¹ÊÙ§
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144511
79162
126313
112714
813211
103165
15694

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3. ·ÒÂàÅ¢·Õèà¸Íà¢Õ¹äÇé

·ÒÂàÅ¢·Õèà¢Õ¹äÇé¤×ÍÍÐäÃ
¢Ñ鹵͹µÑÇÍÂèÒ§
1.à¢Õ¹àÅ¢·Õèµéͧ¡Ò÷ÒÂ12
2. ¤Ù³´éÇ 5 12x5 = 60
3. àÍÒ 6 仺ǡ 60+6 = 66
4. àÍÒ 4 令ٳ66x4 = 264
5. àÍÒ 9 仺ǡ264+9 = 273
6. àÍÒ 5 令ٳ273x5 = 1,365
7. µÑ´àÅ¢ÊͧµÑÇËÅѧ·Ôé§13
8. àÍÒ 1 ä»Åº13-1 = 12

à¾ÃÒЩйÑé¹ 12 ¤×ÍàÅ¢·Õèà¢Õ¹·ÒÂäÇé

Åͧ·´Êͺà¤Åç´ÅѺ¢éÒ§µé¹¡Ñº¨Ó¹Ç¹àÅ¢Í×è¹ÇèÒà»ç¹¨ÃÔ§ËÃ×ÍäÁè




4. Åͧ·ÒÂà¾×è͹´Ù

ÊØªÒµÔºÍ¡¡ÑºÍÒÀóìÇèÒ ÁÕà¡Á·ÒÂàŢʹء â´ÂÊØªÒµÔºÍ¡ãËéÍÒÀóì·Ó´Ñ§¹Õé
¢Ñ鹵͹µÑÇÍÂèÒ§
1.à¢Õ¹àÅ¢ÊÒÁËÅÑ¡¨Ðà»ç¹àÅ¢ÍÐäáçä´é·ÕèàÅ¢·Ñé§ÊÒÁµÑÇäÁè«éӡѹ482
2. ¡ÅѺËÅÑ¡àÅ¢àÊÕÂãËÁè àÍÒµÑÇ˹éÒä»äÇéËÅѧ
àÍÒµÑÇ˹éÒäÇé˹éÒ µÑÇ¡ÅÒ§àËÁ×͹à´ÔÁ
284
3. àÍÒ ¨Ó¹Ç¹·Õè¹éÍÂä»ÅºÍÍ¡¨Ò¡¨Ó¹Ç¹·ÕèÁÒ¡ 482 - 284 = 198
4. ºÍ¡à¾Õ§àÅ¢ËÅÑ¡·éÒ¢ͧ¤ÓµÍº8

àÅ¢·éÒ ¤×Í 8 ÍÒÀóìºÍ¡
ÊØªÒµÔ µÍº ¤ÓµÍº¤×Í 198 áÅÐ͸ԺÒÂËÅÑ¡¡ÒÃÇèÒ
à¸Î¨ÐàËç¹ÇèÒàÅ¢ËÅÑ¡ÃéÍ¡ѺàÅ¢ËÅѡ˹èÇÂàÁ×èͺǡ¡Ñ¹áÅéÇä´éà·èҡѺ 9 àÊÁÍ
´Ñ§¹Ñé¹ àÁ×èÍà¸ÍºÇ¡àÅ¢ËÅÑ¡ÊØ´·éÒÂËÃ×ÍËÅѡ˹èÇÂà·èҡѺ 8 ©Ñ¹¡çàÍÒä»ÅºÍÍ¡¨Ò¡ 9
ä´éà·èҡѺ 1 ¤ÓµÍº¨Ö§»ç¹ 198
ÁÕ¢éÍáÁéÇèÒ ¶éÒàÅ¢ËÅѡ˹èÇÂËÃ×ÍËÅÑ¡ÊØ´·éÒÂà»ç¹ 9 ¨ÐäÁèÁÕàÅ¢ã¹ËÅÑ¡ÃéÍ ¤ÓµÍº¨Ðà»ç¹ "99"

Åͧ·´Åͧ¡ÑºàÅ¢µèÍ仹Õé ´ÙÇèҼŷÕèä´éà»ç¹ÍÂèÒ§äà (" 825 , 125 , 892 , 196")




5. àÅ¢ 4 ÊÕèµÑÇ

¤ÇÒÁʹءÍÂèҧ˹Öè§ã¹¤³ÔµÈÒʵÃì ¤×Í àÃÒ¾ÂÒÂÒÁãªéàÅ¢ 4 à¾Õ§ÊÕèµÑÇá·¹¨Ó¹Ç¹ 1,2,3,4 ... áÅÐàÃ×èÍÂæä»â´ÂÍÒÈÑÂà¤Ã×èͧËÁÒ ºÇ¡ ź ¤Ù³ ËÒà à»ç¹à¤Ã×èͧªèÇ àªè¹
1 = (4 / 4)x(4 / 4)
2 = (4 / 4) + (4 / 4)
3 = (4 + 4 + 4) / 4
4 = 4 x (4 4) + 4
5 = ((4 x 4) + 4)
6 = 4 + ((4 + 4) / 4)
7 = (4 + 4) - (4 / 4)
8 = 4 + 4 + 4 - 4
9 = (4 + 4)+(4 / 4)
10 = (44 - 4) / 4
11 = (44 - 4) / 4
12 = (44 + 4) / 4
13 = 4 + ((4 - .4 ) / .4)
14 = ( 4 x (4 - .4) - .4 )
15 = (4 x 4) - (4 / 4)
16 = 4 + 4 + 4 + 4
17 = (4 x4) + (4 / 4)
18 = (4 x 4)+(4 / .4)
19 = (((4 + 4 ) - .4) / .4)
20 = 4 x (4 + (4 / 4))



6. ·ÒÂà´×͹à¡Ô´áÅÐÍÒÂØ

¢Ñ鹵͹µÑÇÍÂèÒ§
1.¡Ó˹´¤èҢͧà´×͹à¡Ô´â´Â
1 = Á.¤. 2 = ¡.¾. 3 = ÁÕ.¤.
4 = àÁ.Â. 5 = ¾.¤. 6 = ÁÔ.Â.
7 = ¡.¤. 8 = Ê.¤. 9 = ¡.Â.
10 = µ.¤. 11 = ¾.Â. 12 = ¸.¤.
à¡Ô´à´×͹ ÁÕ¹Ò¤Á = 3
2. àÍÒ 2 令ٳ¤èҢͧà´×͹ 3 x 2 = 6
3. àÍÒ 5 仺ǡ 6 + 5 = 11
4. àÍÒ 50 令ٳ11 x 50 = 550
5. àÍÒÍÒÂØä»ºÇ¡àªè¹ÍÒÂØ 36 : 550 +36 = 586
6. àÍÒ 365 ä»Åº586 - 365 = 221
7. àÍÒ 115 仺ǡ221 + 115 = 336

ÊÃØ» ä´éàÅ¢ 3 3 6 à¾ÃÒЩйÑé¹ à¡Ô´à´×͹ ÁÕ¹Ò¤Á ÍÒÂØ 36 »Õ

·èÒ¹ÅͧÊÁÁصÔÍÒÂØáÅÐà´×͹à¡Ô´ãËÁè·´Åͧ´Ù«ÔÇèÒä´éËÃ×ÍäÁè




7. ÇÔ¸Õ¤Ô´ÅÑ´¡Òá¡ÓÅѧÊͧàŢŧ·éÒ´éÇ 5

¢Ñ鹵͹µÑÇÍÂèÒ§ àªè¹ 75 ¡¡ÓÅѧ 2
1.¹ÓàŢ˹éÒ 5 ã¹·Õè¹Õé¤×Í 77
2. ¹Ó 7 ÁÒ¤Ù³¡ÑºàÅ¢ÁÒ¡¡ÇèÒ 7 ÍÂÙè 1 ã¹·Õè¹Õé ¤×Í 87 x 8 = 56
3. ¹Ó¼ÅÅѾ¸ì·Õèä´é¨Ò¡¢éÍ (2) ÁÒàµÔÁ´éÇ 25 5625

à¾ÃÒЩйÑé¹ 75 ¡¡ÓÅѧ 2 = 5625